Evaluate the integral.

${\int}_{0}^{1}{\int}_{0}^{\pi}{\int}_{0}^{\pi}y\cdot \mathrm{cos}\left(z\right)dxdydz$

Brennan Flores
2021-10-25
Answered

Evaluate the integral.

${\int}_{0}^{1}{\int}_{0}^{\pi}{\int}_{0}^{\pi}y\cdot \mathrm{cos}\left(z\right)dxdydz$

You can still ask an expert for help

Demi-Leigh Barrera

Answered 2021-10-26
Author has **97** answers

Step 1

Given that:${\int}_{0}^{1}{\int}_{0}^{\pi}{\int}_{0}^{\pi}y\cdot \mathrm{cos}\left(z\right)dxdydz$

Step 2

As the given integral have constant limits.

So, each integral can be evaluated separately.

Thus,${\int}_{0}^{1}{\int}_{0}^{\pi}{\int}_{0}^{\pi}y\cdot \mathrm{cos}\left(z\right)dxdydz={\int}_{0}^{1}1dx{\int}_{0}^{\pi}ydy{\int}_{0}^{\pi}\mathrm{cos}zdz$

$={\left[x\right]}_{0}^{1}\cdot {\left[\frac{{y}^{2}}{2}\right]}_{0}^{\pi}{\left[\mathrm{sin}z\right]}_{0}^{\pi}$

$=[1-0]\cdot \left[\frac{{\pi}^{2}-{0}^{2}}{2}\right][\mathrm{sin}\pi -\mathrm{sin}0]$

$=1\times \frac{{\pi}^{2}}{2}\times 0$

=0

So,${\int}_{0}^{1}{\int}_{0}^{\pi}{\int}_{0}^{\pi}y\cdot \mathrm{cos}\left(z\right)dxdydz=0$

Given that:

Step 2

As the given integral have constant limits.

So, each integral can be evaluated separately.

Thus,

=0

So,

asked 2020-11-12

Solve the following integral

${\int}_{0}^{4}3x(4-x)dx=32$

${\int}_{0}^{4}x(x-4)dx$

asked 2021-11-05

Evaluate the integral.

$\int \frac{ax}{{x}^{2}-bx}dx$

asked 2022-04-02

Existence of the integral ${\int}_{\alpha}^{\beta}\sqrt{\frac{\mathrm{arctan}h\left(r\right)}{r}}dr$

For$-1\le \alpha \le \beta \le 1$

For

asked 2021-08-14

Convert the indefinite integral to into a definite integral using the interval [0,1], and solve it.

${\int}_{0}^{1}\frac{x}{{({x}^{2}+1)}^{2}}dx$

asked 2021-10-13

Evaluate the integral.

$\int}_{2}^{\mathrm{\infty}}\frac{dx}{{(x+2)}^{2}$

asked 2020-10-19

Evaluate the integral.

$\int {x}^{3}\mathrm{ln}xdx$

asked 2021-09-23

Evaluate the integral.

$\int \frac{tdt}{\sqrt{7-{t}^{2}}}$